Biomedical Engineering Reference
In-Depth Information
The plot of the approximation for the elasticity modulus in the transversal
direction is presented in Fig.
6.6
. For the compression stress in the axial direction
and in the transversal direction the approximation curves are presented respec-
tively in Figs.
6.7
and
6.8
. The ultimate tension stress of the bone tissue is usually
defined as a proportion of the ultimate compression stress, r
i
¼ a
r
i
. There are
several studies on this subject, experimental [
29
-
31
] and numerical [
32
]. The
value of a to be used is not consensual, in the various works available in the
literature a varies between 0.33 and 1.00, as so in this work it will be considered
a ¼ 0
:
5, once it is a conservative value. Although it will be not considered in the
analyses on this topic, the bone shear ultimate stress varies between 49 MPa and
69 MPa [
31
].
6.3 Bone Remodelling Algorithms
Recent experimental investigations shown the existence, in the bone remodelling
process, of a strong correlation between the bone functional adaptations and the
induced stress (or the strain). The strain distribution, the dynamic nature of the
loads and the number of loading cycles seem to be the most significant external
stimuli in the bone remodelling process. However, experimental research has
shown that the bone remodelling and functional adaptation are quite complex and
that both cannot be readily described in detail at the present moment.
In the pursue of explaining the nature of bone remodelling phenomena,
numerous semi empirical laws, phenomenological based mathematical descrip-
tions of the remodelling processes, were developed and proposed. With these semi
empirical laws it is possible to predict the actual stress distribution and stress path
and correlate them with the bone remodelling process. These mathematical for-
mulations consider the bone tissue as a local adaptive material, directly dependent
on the mechanical loading, mostly characterized by the strain or stress tensors, and
aim to numerically predict the local remodelling reactions observed in experiments
through appropriate bone growth laws. The basis of the computational tools
simulating the natural bone adaptation, occurring under known stress states and
changes in geometry and stiffness, are these semi empirical bone growth laws.
Next, a brief overview over bone remodelling theories proposed by several authors
is presented.
6.3.1 Pauwels's Model
Pauwels [
33
] was one of the firsts to suggest a mathematical formulation for the
''Wolff's Law''. In order to ensure a balanced state of bone reabsorption and
deposition, it was presumed the existence of an optimal mechanical stimulus S
n
.
